This paper investigates the local and global well-posedness for the inhomogeneous nonlinear Schr\"{ o}dinger (INLS) equation $iu_{ t} +\Delta u=\lambda \left|x\right|^{ -b} \left|u\right|^{ \sigma } u, u(0)=u_{ 0} \in L^{ 2}(\mathbb R^{ n})$, where $\lambda \in \mathbb C$, $0<b<\min \left\{ 2,{ \rm \; }n\right\}$ and $0<\sigma \le \frac{ 4-2b}{ n} $. We prove the local well-posedness and small data global well-posedness of the INLS equation in the mass-critical case $\sigma =\frac{ 4-2b}{ n} $, which have remained open until now. We also obtain some local well-posedness results in the mass-subcritical case $\sigma <\frac{ 4-2b}{ n} $. In order to obtain the results above, we establish the Strichartz estimates in Lorentz spaces and use the contraction mapping principle based on Strichartz estimates.
Categories: math.AP